Non-stationary signals are produced by systems with attributes that vary with time. The necessity to understand and analyze such non-stationary signals from theoretical, research and diagnostic standpoints emerges in almost every field of biomedical research. However, non-stationary signals are not supported by general signal theory, and suitable mathematical methods to deal with non-stationary signals are quite limited.
Current signal processing methods are generally routed in the theory of linear stationary systems. Accordingly, the non-stationary character of real data is often ignored or the effects presumed to be negligible. Although the corresponding assumptions of the time invariance and linearity provide acceptable results for some applications, many physiological phenomena are highly variable and nonlinear, and therefore do not admit a reasonably accurate linear approximation. Meanwhile, a large body of scientific and clinical evidence indicates that a more comprehensive analysis of major non-stationary physiological signals, such as electrocardiogram (ECG) and electroencephalogram (EEG), is expected to lead to a deeper understanding of the mechanisms and states of various physiological systems under different normal and pathological conditions. In this context, developments of advanced algorithms of non-stationary signal processing emerge as capable approaches to a number of increasingly challenging applications, such as human brain and body imaging, computer implemented health monitoring and EEG based brain-computer interfaces. At the same time, the ongoing advances of computerized biomedical systems, in terms of speed, size, and cost, made the development of sophisticated algorithms practical and cost effective.
Non-stationary signal analysis is not supported by adequate theoretical and computational frameworks, such as in the case of linear stationary systems theory. A heuristic, rather than a mathematically rigorous approach, is generally accepted in the analysis of particular types of physiological signals. For example, when analyzing EEG signals, such signals may be analyzed using the notion of local stationarity (see Barlow, J. S. “Methods of analysis of non-stationary EEGs, with emphasis on segmentation techniques: a comparative review”. J. Clin. Neurophysiol. 1985, vol. 2, pp. 267-304). This notion presumes that, although the signal is inherently non-stationary, within small intervals of time the signal pattern departs only slightly from stationarity (see Priestley, M. B.: “Non-Linear and Non-Stationary Time Series Analysis”, Academic Press, London, 1988). Such quasi-stationary elements are associated with different components of the signal that may be produced by distinct non-linear systems.
So far, the problem of detecting components of non-stationary signals has been approached using data driven and model driven signal processing methodologies.
The advantage of data-driven signal processing methods is their ability to describe dynamics of complex empirical waveforms in detail. One such data-driven method, the empirical mode decomposition (EMD), has been developed by Huang et al. (Huang et al.: “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis”. Proc. R. Soc. London, Ser. A, 1998, vol. 454, pp. 903-995) to decompose on an adaptive basis a non-stationary signal into the sum of “well-behaved” oscillatory functions, termed intrinsic mode functions. In the following developments the EMD has been supported by the Hilbert transform which provided instantaneous frequencies for each intrinsic mode function. Using these measures, the technique describes the system in terms of the instantaneous frequency changes. Yet, the solution is not uniquely defined because it depends on a number of empirical parameters, the choice of which is not supported by strict criteria. Without a firm mathematical foundation of the EMD, this purely empirical and intuitive approach creates a number of difficult methodological problems. For example, the intrinsic mode function is defined only numerically as the output of an iterative algorithm, with no analytical definition. Under these circumstances, the judgments about different aspects of methodology have come from case-by-case comparisons conducted empirically.
The model-driven approach to bio-medical signal processing requires accurate models of the systems involved. A specific aspect of the problem is that physiological signals are the global scale variables produced by multiple microscopic scale sources. Creation of a physical model necessitates a strict identification of relevant microscopic scale elements and the synthesis, on these grounds, the global model. However, the transition from the global to the microscopic scale using these strictly deterministic notions has no unique solution because it is impossible to exactly delineate a distinctive pattern of the cellular and molecular phenomena involved. In the face of such uncertainty, numerous efforts to create models of the cellular machinery that gives rise to the global scale signals are supported by a remarkable variety of heuristic approaches that differ widely not only in physiological and anatomical details of the models, but also in the basic mathematical tools. The extent to which the models are in contradiction is unknown.
Accordingly, there still exists a need, unfulfilled by current methodologies, to effectively identify components of a non-stationary time series signal, so that the corresponding models can be efficaciously utilized for classifying the attributes of the signal.